The reachability problem in lossy counter machines is the best-known ACKERMANN-complete problem and has been used to establish most of the ACKERMANN-hardness statements in the literature. This hides however a complexity gap when the number of counters is fixed. We close this gap and prove F_d-completeness for machines with d counters, which provides the first known uncontrived problems complete for the fast-growing complexity classes at levels 3 < d < omega. We develop for this an approach through antichain factorisations of bad sequences and analysing the length of controlled antichains.
@InProceedings{schmitz:LIPIcs.ICALP.2019.129, author = {Schmitz, Sylvain}, title = {{The Parametric Complexity of Lossy Counter Machines}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {129:1--129:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.129}, URN = {urn:nbn:de:0030-drops-107056}, doi = {10.4230/LIPIcs.ICALP.2019.129}, annote = {Keywords: Counter machine, well-structured system, well-quasi-order, antichain, fast-growing complexity} }
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